‘A damn good fencer’: Maxine Peake as Hamlet at the Royal Exchange. Photograph: Jonathan Keenan

Her face gleams keenly under an immaculate, straw-coloured David Bowie barnet. She wears a dark blue trouser suit that might have been imagined by a fashion-conscious Chairman Mao. She is a stripling prince, almost pre-sexual, who glides, without swagger and without girlishness. Straightaway, Maxine Peake knocks on the head one of the paradoxes of Hamlet. The speeches that come out of the prince’s mouth are about dissolving, yet the person who delivers them has to be the most distinct, intense character on stage.

Peake’s delicate ferocity, her particular mixture of concentration and lightness, ensure that you want to follow her whenever she appears. Anger is her keynote. Her voice is reedy with indignation. The speeches tumble out at high speed, as if she is surprised by her own fervour. Some dark and disturbing notes she does not hit. She is precise rather than cloudy, cutting rather than meditative. She is a damn good fencer.

This is not the apex of Peake’s stage career, but then the bar is high. She will be hard-put to equal the incandescent fervour she mustered when delivering Shelley’s The Masque of Anarchy, directed by Sarah Frankcom at last year’s Manchester international festival. Still, it is a considerable achievement, and the more prominent as hers is the first female Hamlet on a major stage since Frances de la Tour 35 years ago. At first that does not seem surprising, but there is a long, strong tradition of women performing the role, which can hardly be said to demand Schwarzenegger attributes. Sarah Siddons took it on in Manchester in 1777. Victorian actresses, amateur and professional, played the part regularly. Sarah Bernhardt, the first actress to be filmed in the part, declared it should always be performed by a woman.

Tony Howard’s interesting programme note suggests that the dip in female Hamlets in the supposedly feminist 20th century is due to the rise in importance of directors, until recently usually male. That seems right. Put two women in charge of theatres – Sarah Frankcom at the Royal Exchange and Josie Rourke at the Donmar – and there is a sudden burst of parts for women over 40 by cross-gender casting. This autumn, two years after she appeared at the Donmar as Brutus, Harriet Walter will star there as Henry IV.

Frankcom is in effect creating England’s first mainstream feminist theatre. She has done so not only by cross-casting but by her choice of plays, among them The Last Days of Troyand Orlando. Her production of Hamlet has a Player Queen (the powerful and mellifluous Claire Benedict), female gravediggers (Michelle Butterly is a jaunty scouser) and a knowing biker of a Rosencrantz (Jodie McNee). Gillian Bevan makes her courtly adviser (“Polonia”) a fusspot, continually tweaking her dark suit into place and accompanying her orotund phrases with redundant flourishes of her hands. She is highly entertaining, though not a major contributor to the sinisterness of Elsinore. These gender switches may unsettle for a moment but they do not distort the play. At least, not unless you think that “to be, or not to be” can only refer to people with penises.

Amanda Stoodley’s design provides some distracting moments. Ophelia’s grave is a jumble of garments. Why? Does this refer to apparel oft proclaiming the man? Yet it supplies a marvellous entry for the Ghost. John Shrapnel, doubling as an impressive, stentorian Claudius, is heralded by the lowering of a forest of lightbulbs which, accompanied by a clanging chime, glow and fade to the bewilderment of watchers. In general the staging is simple, in line with a stripped-down text that has no Fortinbras in it and therefore little political content. As someone who never holds her breath for Fortinbras to appear, I find this more a theoretical than actual diminishment. The important thing is that Hamlet should take you into its speech, and so into the archaeology of everyday conversation. Which this production does, clearly, energetically though not superlatively.